INTERPRETING
REGRESSION
COEFFICIENTS
OUTLINE
1.
2.
3.
4.
5.
6.
7.

Back to Basics
Form: The Regression Equation
Strength: PRE and r2
The Correlation Coefficien...
BACK TO BASIC CONCEPTS
PRE = (E1 – E2)/E1 = 1 – E2/E1
E1 = Σ(Y – Y)2
Rule for “predicting” values of Y, given knowledge of...
E2 = Σ (Yi – Ŷ)2
that is, sum of squared differences between observed
values of Y and predicted values of Y (values of Y a...
STRENGTH OF ASSOCIATION
Symbol = r2 = PRE = (E1 – E2)/E1
= (total variance – unexplained variance)/total variance
Varies f...
FOCUSING ON FORM
As given by equation Ŷi = a + bXi

Constant a = intercept = predicted value of Y when X = 0
Coefficient b...
Linear Regression Equation
THE CORRELATION COEFFICIENT
Symbol = r
Summary statement of form (from sign) and indirect
statement of strength
r = square...
ON THE CORRELATION COEFFICIENT r
Analogous to slope b (with removal of intercept a)
The “standardized regression coefficie...
LOOKING AHEAD:
MEASURING SIGNIFICANCE
1. Testing the null hypothesis:
F = r2(n-2)/(1-r2)
2. Standard errors and confidence...
Figure 1. Cycles of Political Change in Latin America, 1900-2000
19
18
17
16
15
14
13
12
Number

11
Semi-Democracy
Oligarc...
Coefficients for Regression of N Electoral Democracies (Y)
on Change Over Time (X):
a = -1.427
b = +.126
r = + .883
r2 = ....
Scatterplot: N Democracies by Year
Interpreting the Equation

• N democracies = - 1.427 + .126 year
• intercept = nonsense, but allows calculation of
year th...
Example 2: Wine and Heart Disease
Data in Lectures 5-6
X = per capita annual consumption of alcohol from
wine, in liters
Y...
Poli30 session14 2008
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Poli30 session14 2008

  1. 1. INTERPRETING REGRESSION COEFFICIENTS
  2. 2. OUTLINE 1. 2. 3. 4. 5. 6. 7. Back to Basics Form: The Regression Equation Strength: PRE and r2 The Correlation Coefficient r Significance: Looking Ahead Example 1: Democracy in Latin America Example 2: Wine Consumption and Heart Disease
  3. 3. BACK TO BASIC CONCEPTS PRE = (E1 – E2)/E1 = 1 – E2/E1 E1 = Σ(Y – Y)2 Rule for “predicting” values of Y, given knowledge of X: Yhati = a + bXi
  4. 4. E2 = Σ (Yi – Ŷ)2 that is, sum of squared differences between observed values of Y and predicted values of Y (values of Y as “predicted” by the regression equation) Thus the elements of PRE.
  5. 5. STRENGTH OF ASSOCIATION Symbol = r2 = PRE = (E1 – E2)/E1 = (total variance – unexplained variance)/total variance Varies from 0 to 1 Some back-of-the-envelope thresholds: 0.10, 0.30, 0.50+
  6. 6. FOCUSING ON FORM As given by equation Ŷi = a + bXi Constant a = intercept = predicted value of Y when X = 0 Coefficient b = slope = average change in Y for change in X • Magnitude (large or small) • Sign (positive or negative) • Key to much interpretation
  7. 7. Linear Regression Equation
  8. 8. THE CORRELATION COEFFICIENT Symbol = r Summary statement of form (from sign) and indirect statement of strength r = square root of r2, varies from –1 to +1 subject to over-interpretation useful for preliminary assessment of association Symmetrical no matter which variable is X and which is Y (note: slope b is not symmetrical)
  9. 9. ON THE CORRELATION COEFFICIENT r Analogous to slope b (with removal of intercept a) The “standardized regression coefficient,” or beta weight: β= b (stand.dev. X/stand.dev. Y) employs slope, values, and dispersion of variables thus a “standardized” slope Question: How much action on Y do you get from X? In bivariate (or “simple”) regression, β = r
  10. 10. LOOKING AHEAD: MEASURING SIGNIFICANCE 1. Testing the null hypothesis: F = r2(n-2)/(1-r2) 2. Standard errors and confidence intervals: Dependent on desired significance level Bands around the regression line 95% confidence interval ±1.96 x SE
  11. 11. Figure 1. Cycles of Political Change in Latin America, 1900-2000 19 18 17 16 15 14 13 12 Number 11 Semi-Democracy Oligarchy Democracy 10 9 8 7 6 5 4 3 2 1 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Year
  12. 12. Coefficients for Regression of N Electoral Democracies (Y) on Change Over Time (X): a = -1.427 b = +.126 r = + .883 r2 = .780, Adjusted r2 = .777 Standard error of slope = .0067 95% confidence interval for slope = (.0067)x1.96 = ± .0013 setting confidence bands at .113 and .140 F for equation = 350.91, p < 0.000
  13. 13. Scatterplot: N Democracies by Year
  14. 14. Interpreting the Equation • N democracies = - 1.427 + .126 year • intercept = nonsense, but allows calculation of year that predicted value of Y would be zero, in this case 1910 • slope = +.126 so, one additional democracy every eight years • and by 2000, total 11-12 democracies • PRE = .777
  15. 15. Example 2: Wine and Heart Disease Data in Lectures 5-6 X = per capita annual consumption of alcohol from wine, in liters Y = deaths from heart disease, per 100,000 people Equation: Ŷ = 260.6 - 22.97 X r = - 0.843 What’s the interpretation?
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